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I am applying for a PhD in Math at US grad schools mainly because they allow you one or more years of coursework to figure out your research interests and potential advisors. Are there any other countries that allow you this opportunity outside of the US? I have an MSc that was purely coursework based without a thesis component and while I'm certain of the subfield I'd like to work in, I still need more time to figure out the specific topics of my research and create a research proposal.

From what I've read here and here Europe seems to be on the opposite side of this spectrum. Are there any (preferably English speaking) countries that are closer to the US in this regard?

ZSMJ
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    This isn't a duplicate, but see: https://academia.stackexchange.com/q/176908/75368, though it is more about the application process. – Buffy Oct 21 '22 at 14:34
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    @Buffy Thanks, I've gone through it already and mentioned it in my post. – ZSMJ Oct 21 '22 at 14:36
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    It might vary even between universities. That said, I am doing mine in Israel. I had to choose an advisor when applying (but this is for my university, I know that e.g. in the Weizmann institute, they do rotations in the 1st year) but I only had to submit a research proposal after 18 months into the PhD. – aqua Oct 24 '22 at 08:20
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    I don't know if there is another country that has this or not, but as for the US, some universities give you a lot more flexibility than others. Make sure to contact current PhD students and ask before applying/accepting. Don't assume you will have all the freedom you want just because the university is in the US. – stochastic Oct 25 '22 at 09:58

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This has as much to do with the specific project and its circumstances as it does with the country.

In my own country, the UK, some projects begin life clearly defined: "The appointed student will use methods x,y, and z to determine whether A and B are affected by C". That isn't to say that the project might not change and evolve over 3 years, but it is clearly set out to begin with. This is perhaps more common in STEM subjects, where one or more PhDs might form part of the research for a larger project.

Other projects are on more general topics, where it is expected that the student will narrow their interests and more clearly define the project during the first 6-9 months.

Something to note is that no time is "allowed" for this - it eats into the time available (usually 3 or 3.5 years). So in some senses this is less flexible than the US approach.

Flyto
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US grad schools mainly because they allow you one or more years of coursework

They require coursework. "Allow" is the wrong word.

to figure out your research interests

That's not the purpose of the coursework. You may be allowed to take time to decide your research topic, but that does not mean you should. The most successful PhD students will have a plan for completing their PhD before starting it.

Are there any other countries that allow you this opportunity outside of the US?

It's normal for research projects to evolve over time. I do not think there is any PhD program that would be exempt from that.

The flexibility available to a PhD student is based primarily on funding, not the country.

Anonymous Physicist
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    I disagree, at least in principle: at my R1 state univ in the U.S., in mathematics, we do promote the idea that, of course, getting more information from a year or two of forward-looking informative coursework, before deciding on a thesis direction, is very wise. The coursework is not meant to be "filtering"... the word "requirement" is misleading... – paul garrett Oct 24 '22 at 01:03
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    @paulgarrett, my experience is the same as yours, however, I've recently learned (discussion on meta) that math may be an outlier on several dimensions of study and research. In particular, serious (publishable) undergraduate research seems to be rare in math, but much less so in other disciplines. So, this answer doesn't resonate for math, but it probably does for others. The OP is, of course, asking about math. – Buffy Oct 24 '22 at 12:36
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    I don't want to downvote your answer just because I disagree, but nothing you said resonates with me. I studied in the US and work in Europe. The US system is by far more flexible than the European system. Universities do "allow" you to take courses beyond their requirements (which in some cases is minimal). They do encourage you to figure out your interests by taking courses and even by rotating between research groups. And in the European system, the time is tight and the research project is more or less fixed at least by topic. – stochastic Oct 25 '22 at 10:04
  • @stochastic I do not believe you that the project is fixed by topic. The project is fixed by which topics have funding. If there's funding, the student can switch projects, though it may be extremely inconvenient. Perhaps they have to change universities and start over. PhD students are not prisoners in Europe. – Anonymous Physicist Oct 26 '22 at 01:24